Existence of mild solutions for semilinear equation of evolution

Anna Karczewska; Stanisław Wędrychowicz

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 4, page 695-706
  • ISSN: 0010-2628

Abstract

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The aim of this paper is to give an existence theorem for a semilinear equation of evolution in the case when the generator of semigroup of operators depends on time parameter. The paper is a generalization of [2]. Basing on the notion of a measure of noncompactness in Banach space, we prove the existence of mild solutions of the equation considered. Additionally, the applicability of the results obtained to control theory is also shown. The main theorem of the paper allows to characterize the set of controls providing solutions of the system considered. Moreover, the application of the main theorem for elliptic equations is given.

How to cite

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Karczewska, Anna, and Wędrychowicz, Stanisław. "Existence of mild solutions for semilinear equation of evolution." Commentationes Mathematicae Universitatis Carolinae 37.4 (1996): 695-706. <http://eudml.org/doc/247879>.

@article{Karczewska1996,
abstract = {The aim of this paper is to give an existence theorem for a semilinear equation of evolution in the case when the generator of semigroup of operators depends on time parameter. The paper is a generalization of [2]. Basing on the notion of a measure of noncompactness in Banach space, we prove the existence of mild solutions of the equation considered. Additionally, the applicability of the results obtained to control theory is also shown. The main theorem of the paper allows to characterize the set of controls providing solutions of the system considered. Moreover, the application of the main theorem for elliptic equations is given.},
author = {Karczewska, Anna, Wędrychowicz, Stanisław},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semilinear equation of evolution; mild solutions; measure of noncompactness; sublinear measure; semilinear evolution equations; mild solutions; sublinear measure of noncompactness},
language = {eng},
number = {4},
pages = {695-706},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Existence of mild solutions for semilinear equation of evolution},
url = {http://eudml.org/doc/247879},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Karczewska, Anna
AU - Wędrychowicz, Stanisław
TI - Existence of mild solutions for semilinear equation of evolution
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 4
SP - 695
EP - 706
AB - The aim of this paper is to give an existence theorem for a semilinear equation of evolution in the case when the generator of semigroup of operators depends on time parameter. The paper is a generalization of [2]. Basing on the notion of a measure of noncompactness in Banach space, we prove the existence of mild solutions of the equation considered. Additionally, the applicability of the results obtained to control theory is also shown. The main theorem of the paper allows to characterize the set of controls providing solutions of the system considered. Moreover, the application of the main theorem for elliptic equations is given.
LA - eng
KW - semilinear equation of evolution; mild solutions; measure of noncompactness; sublinear measure; semilinear evolution equations; mild solutions; sublinear measure of noncompactness
UR - http://eudml.org/doc/247879
ER -

References

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  1. Banaś J., Goebel K., Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math. Marcel Dekker New York and Basel (1980). (1980) MR0591679
  2. Banaś J., Hajnosz A., Wȩdrychowicz S., Some generalization of Szufla's theorem for ordinary differential equations in Banach space, Bull. Pol. Acad. Sci., Math. XXIX , No 9-10 (1981), 459-464. (1981) MR0646334
  3. Coddington E.A., Levinson N., Theory of Ordinary Differential Equations, Mc Graw-Hill New York-Toronto-London (1955). (1955) Zbl0064.33002MR0069338
  4. Friedman A., Partial Differential Equations, Krieger Publishing Company Huntington New York (1976). (1976) MR0454266
  5. Kato T., Quasi-linear equations of evolution with application to partial differential equations, Lect. Notes Math. Springer Verlag 448 (1975), 25-70. (1975) MR0407477
  6. Kuratowski K., Topologie, Volume II PWN Warszawa (1961). (1961) Zbl0102.37602MR0133124
  7. Pazy A., A class of semi-linear equations of evolution, Isr. J. Math. 20 (1975), 22-36. (1975) Zbl0305.47022MR0374996
  8. Pazy A., Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York (1983). (1983) MR0710486
  9. Rolewicz S., Functional Analysis and Control Theory Linear Systems, Reidel Publishing Company Dordrecht (1987). (1987) Zbl0633.93002MR0920371

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